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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
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<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
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<li><a href="sec_1.html" data-scroll="sec_1" class="internal">Homogeneous equations with constant coefficient</a></li>
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<li><a href="sec_3.html" data-scroll="sec_3" class="internal">Linear Independence and Wronskian</a></li>
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<li><a href="sec_1.html" data-scroll="sec_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec_2.html" data-scroll="sec_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec_3.html" data-scroll="sec_3" class="internal">The Method of Undetermined Coefficients</a></li>
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<li><a href="sec_1.html" data-scroll="sec_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec_2.html" data-scroll="sec_2" class="internal">Introduction</a></li>
<li><a href="sec_3.html" data-scroll="sec_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec_4.html" data-scroll="sec_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec_5.html" data-scroll="sec_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec_1.html" data-scroll="sec_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec_2.html" data-scroll="sec_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec_3.html" data-scroll="sec_3" class="internal">Complex Eigenvalues</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec_1.html" data-scroll="sec_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec_2.html" data-scroll="sec_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec_3.html" data-scroll="sec_3" class="internal">Fourier Series</a></li>
<li><a href="sec_4.html" data-scroll="sec_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec_5.html" data-scroll="sec_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec_6.html" data-scroll="sec_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec_7.html" data-scroll="sec_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec_8.html" data-scroll="sec_8" class="active">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
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<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="internal">Laplace transform for PDE (heat equation)</a></li>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec_8"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">7.8</span> <span class="title">Other Heat Conduction Problems</span>
</h2>
<p id="p-416"><dfn class="terminology">Neumann Boundary Condition.</dfn> We now consider a set of new BCs: (insulated)</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u_x(0,t)=0,\quad u_x(L,t)=0.
\end{equation*}
</div>
<p class="continuation">This means that both ends are insulated, and no heat flows through. Following the same setting, we get the eigenvalue problem for <span class="process-math">\(X(x)\)</span> as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
X''+\lambda X=0,\quad X'(0)=X'(L)=0,
\end{equation*}
</div>
<p class="continuation">From Example 2 in Section <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "nbcex" missing or not unique]</code>, we have only nonnegative eigenvalues <span class="process-math">\(\lambda_n\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\lambda_n={\left(\frac{n\pi}{L}\right)^2},\quad {X_n(x)}={\cos\frac{n\pi x}{L}},\quad n=\textcolor{red}{0},1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">The solution for <span class="process-math">\(T(t)\)</span> remains the same</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
T_n(t)=C_n\cdot\exp\left[-\left(\frac{n\pi\alpha}{L}\right)^2t\right], \quad n=\textcolor{red}{0},1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">which leads to the formal solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=C_0 + \sum_{n=1}^{\infty}X_n(x)T_n(t)=C_0 + \sum_{n=1}^{\infty}C_n e^{-\frac{n^2\pi^2\alpha^2}{L^2}t} \cos \frac{n\pi x}{L},
\end{equation*}
</div>
<p class="continuation">Finally, by fitting in the initial condition, <span class="process-math">\(C_n\)</span> can be determined as the Fourier cosine coefficient for the even half-range expansion of <span class="process-math">\(f(x)\text{,}\)</span> i.e.,</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_0=\frac{1}{2}\frac{2}{L}\int_{0}^{L} f(x)~\textrm{d}x=\frac{1}{L}\int_{0}^{L} f(x)~\textrm{d}x,\qquad C_n=\frac{2}{L}\int_{0}^{L} f(x)\cos \frac{n\pi x}{L}~\textrm{d}x, \quad n=1,2,3,\cdots.
\end{equation*}
</div>
<ul id="p-417" class="disc">
<li id="li-62"><p id="p-418">Harmonic oscillation in <span class="process-math">\(x\text{,}\)</span> exponential decay in <span class="process-math">\(t\text{.}\)</span></p></li>
<li id="li-63"><p id="p-419">exponential decay in <span class="process-math">\(t\text{,}\)</span> except the term <span class="process-math">\(C_0\text{.}\)</span> Decay faster for larger <span class="process-math">\(n\text{.}\)</span></p></li>
<li id="li-64"><p id="p-420">As <span class="process-math">\(t\to\infty\text{,}\)</span> <span class="process-math">\(u(x, t) \to C_0\text{,}\)</span> which is the average of <span class="process-math">\(f(x)\)</span> (initial temperature). This is reasonable because the bar is insulated.</p></li>
</ul>
<p id="p-421">As <span class="process-math">\(t \to\infty\text{,}\)</span> solution does not change in time anymore, as it reaches a steady state, say <span class="process-math">\(U(x)\text{.}\)</span> Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U_t = 0\quad\to\quad U_{xx}=U_t = 0\quad\to\quad U(x) = Ax + B,
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(A\text{,}\)</span> <span class="process-math">\(B\)</span> are determined by boundary conditions.</p>
<p id="p-422">(1) If <span class="process-math">\(u(0,t) = a\text{,}\)</span> <span class="process-math">\(u(L,t) = b\text{,}\)</span> then <span class="process-math">\(U(0) = a\text{,}\)</span> <span class="process-math">\(U(L) = b\text{,}\)</span> we get</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U(x) = a + \frac{b-a}{L}x.
\end{equation*}
</div>
<p class="continuation">(2) If <span class="process-math">\(u(0,t) = a\text{,}\)</span> <span class="process-math">\(u_x(L,t) = 0\text{,}\)</span> then <span class="process-math">\(U(0) = a\text{,}\)</span> <span class="process-math">\(U'(L) = 0\text{,}\)</span> we get <span class="process-math">\(U(x) = a\text{.}\)</span>(3) If <span class="process-math">\(u(0,t) = a\text{,}\)</span> <span class="process-math">\(u_x(L,t) = b\text{,}\)</span> then <span class="process-math">\(U(0) = a\text{,}\)</span> <span class="process-math">\(U'(L) = b\text{,}\)</span> we get <span class="process-math">\(U(x) = bx + a\text{.}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="" id="p-423">
\begin{equation*}
u_t = \alpha^2 u_{xx},\quad u(0,t) = a, ~~u(L,t) = b.
\end{equation*}
</div>
<p class="continuation">We know that the steady state is <span class="process-math">\(U(x) = a + \frac{b-a}{L}x\text{.}\)</span> Now define a new variable</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x, t) = u(x, t) - U (x).
\end{equation*}
</div>
<p class="continuation">Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w_t=u_t,\quad w_x=u_x-U'(x),\quad w(L,t)=u(L,t)-U(L)=b-b=0
\end{equation*}
</div>
<p class="continuation">which are homogeneous. Then, one can find the solution for <span class="process-math">\(w\)</span> by the standard separation of variables and Fourier series. Once this is done, one can go back to <span class="process-math">\(u\)</span> by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=w(x,t)+U(x).
\end{equation*}
</div>
<p id="p-424">Find the solution to the heat equation <span class="process-math">\(u_t = u_{xx}\)</span> with the following BCs</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(0,t)=2,\quad u(1,t)=4,
\end{equation*}
</div>
<p class="continuation">and IC</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,0)=2+2x-\sin \pi x-3\sin 3\pi x.
\end{equation*}
</div>
<p id="p-425">Denote the steady state solution as <span class="process-math">\(U(x)\text{,}\)</span> then it satisfies the following two-point boundary value problem</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U''=0,\quad U(0)=2,\quad U(1)=4,
\end{equation*}
</div>
<p class="continuation">which gives the solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
U(x) = 2 + 2x.
\end{equation*}
</div>
<p class="continuation">Let <span class="process-math">\(w(x,t)\)</span> be the solution of the heat equation with homogeneous boundary condition. Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w_t=w_{xx},\quad w(0,t)=w(1,t)=0,\quad w(x,0)=u(x,0)-U(x)=-\sin\pi x-3\sin 3\pi x.
\end{equation*}
</div>
<p class="continuation">The formal solution for <span class="process-math">\(w\)</span> is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x,t)=\sum_{n=1}^{\infty}C_n e^{-n^2\pi^2 t}\sin(n\pi x),\qquad n=1,2,3,\cdots
\end{equation*}
</div>
<p class="continuation">Here <span class="process-math">\(C_n\)</span> are Fourier coefficients of the initial data <span class="process-math">\(w(x,0)\text{.}\)</span> We find only two coefficients that are not 0, namely</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
C_1=-1,\quad C_3=-3.
\end{equation*}
</div>
<p class="continuation">i.e.</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
w(x,t)=-e^{-\pi^2 t}\sin(\pi x) - 3e^{-9\pi^2 t}\sin (3\pi x).
\end{equation*}
</div>
<p class="continuation">Finally combine <span class="process-math">\(w(x,t)\)</span> together with <span class="process-math">\(U(x)\text{,}\)</span> we get the solution</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
u(x,t)=w(x,t)+U(x)=2+2x-e^{-\pi^2 t}\sin(\pi x) - 3e^{-9\pi^2 t}\sin (3\pi x).
\end{equation*}
</div>
<p id="p-426">This separation of variables technique could be applied to a more general class of PDEs. It is not difficult to check whether an equation is separable. After separating the variables, one needs to fit in the boundary condition. Other than the heat conduct problem, we can use the separation of variable to solve a 1-D Wave equation or a 2D Laplace equation. You may find more details in the textbook.</p></section></div></main>
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